Download design and analysis of axial-flux coreless permanent magnet disk

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DESIGN AND ANALYSIS OF AXIAL-FLUX CORELESS
PERMANENT MAGNET DISK GENERATOR
Łukasz DR ZIKOWSKI
Włodzimierz KOCZARA
Institute of Control and Industrial Electronics
Warsaw University of Technology, Warsaw, Poland
e-mail: [email protected], [email protected]
Abstract: Magnets are an important part of
modern electric machines. Thanks to them motors
become lighter and reliable because of brushless
technology. This article presents simple and easy
to manufacture design of permanent magnet
generator based on coreless windings. This axialflux machine seems to be very interesting for low
speed power generation systems such as wind and
water turbines. There is also shown an example of
basic calculations based on equivalent magnetic
circuit. The analysis presents flux dependence on
several parameters such as: magnet’s grade and
size in comparison with coil and air-gap
dimensions. Second part of the article
concentrates on simulation results of Finite
Element Method analysis (FEM) that clearly
shows the flux distribution for different magnet
shapes – trapezoidal and rectangular. Finally
there is presented a description of a 20kW
prototype of PMSG based on rectangular magnets
which contains mechanical design and a
experimental results.
Keywords: Permanent magnet, FEM, disk
generator, coreless, ironless, air-core, axial-flux,
PMSM, PMSG, AFPM, air-gap, magnetic circuit,
coil, windings, rotor.
1. Introduction
In electric generators magnets function as
transducers, transforming mechanical energy to
electrical energy without any permanent loss of
their own energy. That means the better magnet
the better transducer and finally the better whole
device. That is why, it is so important to learn
basics of magnets before designing the machine.
Nowadays, Neodymium Iron Boron (NdFeB)
magnets are widely known as the best one [1][2].
This modern magnetic material is easily available
on the market in different grades and shapes.
Thanks to them high efficiency electric machines
can be designed. Permanent magnet electric
generators are needed in renewable energy
sources. They can provide power even during
electrical network failure because of built-in
permanent self-excitation.
2. Parameters of neodymium
magnets
Magnetic material can be described by the B(H)
curve. Second quadrant of this characteristic is
called demagnetization curve and is useful for
magnetic calculation. The point where this
characteristic cross H axis is called Coercivity (Hc)
which is a value of external magnetic field
required to reduce magnetization of this material
to zero. Second important parameter is remanence
(Br) which is residual flux density B. Value of Br
tells about the magnetization of material when
external magnetic field (H) is removed. Being
based on those two components it is possible to
determine demagnetization curve. This function is
almost linear for neodymium magnets and can be
approximate by the equation (1).
B( H ) =
Br
H + Br
Hc
(1)
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Having this equation it is easy to determine a
maximum energy product (BH)max. This parameter
of permanent magnet is used by the manufacturers
to specify the grade of the magnet and can be
calculated by the equation (2).
( BH ) max = 0,25 ⋅ Br H c
(2)
The higher energy density the smaller volume of
magnet is needed to obtain the same flux density
in the air-gap. Grade of permanent magnet is often
shown in CGS (Centimeter-Gram-Second) system
of units (MGOe) and it functions on the market as
a symbol of N (e.g. N45). Units conversion factors
are presented in table (1).
Table 1: Units conversion factors.
System
CGS
SI
of units
Flux
density
10 −4 T
1 Gs
Figure 1: Simplified magnetic circuit.
It is resistant for short time overload because of
air-core which is free of saturation. The biggest
drawback is a need of use stronger magnets to
obtain the same power like in conventional ironcore machine which is the reason of higher
reluctance of magnetic circuit. Besides some
sensorless control methods cannot be applied in
this topology [6]. For analytical discussion,
magnetic circuit of this machine can be simplified
to four magnets – two on first rotor disk and other
two on the second disk. That is shown in the
picture (1).
Coil
Magnetic
field
1 Oe
Energy
density
(grade)
1 MGOe
1
4π ⋅ 10 −3
1
4π ⋅ 10
−2
≈ 79,58 A/m
Magnet
≈ 7,96 kJ/m3
3. Machine topology and mechanical design
of magnetic circuit
Presented device is a 12 pair poles synchronous
machine. Double rotor is equipped with 24
magnets per disk. Air-core three phase windings
are placed between magnets (Fig. 2). This
topology has several advantages, e.g. there is no
starting and cogging torque what is very important
especially for wind power systems. Moreover, it is
cheap, easy to manufacture, and can be produced
in the range of single watts up to hundreds of kilo
watts of power in multi disk operation
[1][2][3][4][5].
Figure 2: Isometric view of mechanical topology.
4. Analytical calculations based on
equivalent magnetic circuit
Magnetic circuits are very similar to electric
circuits, so it is possible to use the same analytical
equations for both. Especially, in presented case,
where the core is free of iron, this transformation
is proper for whole range of magnetic fields.
Analytical modeling is enough to calculate
approximated flux density in the air-gap, which is
the base of father voltage and power calculations.
Equivalent magnetic circuit (Fig. 3) can be
described by following equation (3).
Bm =
φ
S
=
Θ
2Θ PM
=
(3)
S ⋅ Rm S ⋅ ( RAG + 2 RPM )
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where: Bm − maximum of flux density in the airgap, φ − the flux that goes through the area S, RAG
and RPM − the reluctances of air-gap and
permanent magnet, Θ − total magnetomotive
force.
when the air-gap length is high. In that case it can
be assumed that flux density shape is sinusoidal.
That means that flux distribution may be expressed
as a function of sin(x) (7).
B( x) = Bm sin
π
x
τ
(7)
Pole pitch τ is a distance between magnets placed
on the same rotor disk. In presented topology pole
pitch is a function of rotor radius r. There is 24
magnets on each disk so that means that angular
pole pitch is 360°/24 = 15°. For analysis it is
needed to have linear pole pitch in [mm] what can
be calculated by following equation (8).
τ = 2r sin(7.5°)
Figure 3: Equivalent magnetic circuit.
Magnetomotive force depends on two parameters
– magnet thickness g and magnet coercivity Hc. It
is given by formula (4).
Θ PM = g ⋅ H c
(4)
Reluctance of material can be easily calculated by
equation (5). Value of reluctance depends on
material permeability µ and its dimensions −
length l and area S that flux is going through.
R=
l
µ ⋅S
(8)
Calculations of electromotive force depend on coil
dimensions. Because of coreless topology, wires
are distributed along the air-gap. That means that
every wire of the coil works in different
conditions. Some of them are in the strong
magnetic field while other are in the edge of
magnet where the flux density is much lower. This
must be taken into account while calculation EMF.
The idea is to calculate average value of flux
density at the area where the coil is. This can be
expressed by integral equation (9).
(5)
Finally the maximum flux density between
magnets is described by the expression (6), where
µ0 is air permeability that equals 4π ⋅ 10− 7
δ is air-gap length.
Bm =
2 gµ0 Br H c
Brδ + 2 gµ0 H c
H
and
m
(6)
This is only approximated value of maximum flux
density because of leakage flux which cannot be
easily calculated. Leakage flux mainly depends on
air-gap and magnets dimension. The worst case is
Figure 4: Coil dimensions.
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2πr ⋅ n 1
E ( x) = k
lz
60
c
x +1 2 c
x +1 2 c
π
π
Bm sin
x dx −
Bm sin
( x + d ) dx
τ
τ
x −1 2 c
x −1 2 c
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(9)
where:
k – number of coils per phase; r – radius; n – rotor rotation speed [rpm];
l – length of conducting wire; z – number of turns; c – coil thickness (Fig. 4);
d – coil diameter (Fig. 4); x – actual position [mm]; E – electromotive force.
5. Air gap size optimization
Equation (6) describes dependence between flux
density and air-gap size. This relation is strongly
nonlinear. Moreover, several other parameters
must be taken into account while calculating
output power of the machine. Increasing air-gap,
value of flux density is falling down while number
of turns is rising. That has an influence on total
impedance. Relation between impedance and
number of turns in also nonlinear, because
inductance of the coil depends on the square of
number of turns. In total output power as a
function of air-gap has a maximum shown in
figure (5).
pole pitch as a function of rotor radius (8). In
similar way, diameter of coil (Fig. 4) can be
calculated. In order to calculate, the best relation
between pole pitch and coil diameter equation (9)
can be used. It consists of the difference of two
sinusoidal functions. This formula will reach
maximum only if those two sinuses are shifted in
180°. It can be achieved when coil width d is equal
to pole pitch τ. Second parameter of the coil is its
thickness c shown in figure (4). In order to keep
diameter of the coil equal to pole pitch, the
thickness of the coil cannot be increase more than
33% of pole pitch because of geometrical
relations. It must be taken into account that this is
three-phase system, so the distance between coils
must be equal 133% of the pole pitch.
Figure 5: Approximated power of the generator.
For this machine optimum air-gap size is about 35
mm. Looking on this characteristic, it can be seen
that the output power is almost constant between
33 and 38 mm of the air-gap. For economic
reasons, the lower size has been chosen (33 mm)
in the prototype, because increasing air-gap in this
range has no significant influence on total output
power.
6. Coil and magnet size
optimization
Coil and magnet size optimization are based on
geometrical analysis. For given number of pair
poles, which is 12, it is easy to calculate linear
Figure 6: Approximated power of the generator.
Apparently in this case, there is not a maximum
power (Fig. 6). It is possible to increase coil
thickness to 50% of pole pitch while decreasing
diameter from 100% to 83,3% of pole pitch.
Decreasing of the diameter has a lower influence
than increasing the number of turns. In total
maximum power can be achieved in this case.
All those analysis have been made under
assumption that the shape of flux density is
sinusoidal. In order to increase maximum power,
the width of the magnet can be also changed.
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Figure 7: Comparison of flux densities.
Figure 8: Comparison of line voltages.
Figure 9: Comparison of line-to-line voltages.
The influence of widening the magnet on flux
density shape cannot be easily calculated, but can
be approximated. Maximum width of the magnet
is equal to pole pitch. In that case, there is no free
space between the magnets, so the shape of flux
density should be more trapezoidal. Finite element
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analysis and intuition says that the wider magnet
the more trapezoidal shape of flux density.
Pictures (7), (8), (9) are based on analytical
calculations and they illustrate the phase voltage
and line-to-line voltage for sinusoidal and
trapezoidal flux density (Fig. 7). While widening
the magnets, the third harmonic appears in line
voltage (Fig. 8). The final result is that the total
output voltage has been increased of 20% (Fig. 9).
7. Flux density analysis for
different shapes of PM
Being based on the results of widening the magnet,
next optimization step can be proposed.
Considering that the linear pole pitch depends on
the rotor radius (8) trapezoidal shape of the
magnet can be applied in order to increase output
voltage (Fig. 2).
follows that distance between two magnets should
be equal to the thickness of the magnet (Fig. 10).
8. Experimental results
Presented topology of three-phase permanent
magnet synchronous machine has been already
implemented. All the mechanics has been designed
in three-dimensional software. To achieve proper
assembly of windings in order to keep very small
air-gap between magnets and coils special tools
and equipment has been designed and produced.
Double disk rotor is equipped with double rolled
spherical bearings to obtain long term life time.
Perforated case
a)
b)
Figure 11: Picture of PMSG.
Figure 10: Flux density for different linear pole
pitch
The simple two-dimensional finite element
analysis has been made for different shapes of
magnets. While widening magnet up to 100% of
pole pitch some part of magnetic field is lost. To
avoid this phenomenon free space between
magnets should be designed. From FEM analysis
Figure 12: Line-to-line EMF.
Case of the machine has been cut out from the
10mm iron sheet and covered thin perforated
aluminum sheet to obtain proper cooling (Fig. 11).
Line-to-line voltage is a sinusoidal wave form with
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only 1,6% of 5th harmonic (Fig. 12). Detailed
specifications are shown below (Table 2).
Table 2: Specification of PMSG.
speed / frequency
URMS line-to-line
unloaded / fully
loaded
URMS line
unloaded / fully
loaded
250 rpm / 50 Hz
140
180 rpm
120
150 rpm
100
100 rpm
80
380 V / 353 V
60
60 rpm
40
220V / 204 V
20
0
IRMS phase
15,0 A
Power
9,1 kW
Weight
190 kg
Dimensions
818/780/176 mm
0
Line-to-line
voltage
3
6
9
12
RMS Phase current [A]
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Figure 14: PMSG output characteristic
In laboratory condition the load test has been done.
There have been measured true RMS values of
voltages and currents for different speeds. There
was take into account only resistive load.
Phase
voltage
RMS Phase voltage [V]
160
Speed
60 rpm
Phase
current
Figure 13: Example oscillograph of current and
voltages during resistive load
Based on those oscillographs the output
characteristic can be drawn. (Fig. 14) Generator
has a strong output characteristic. The voltage
drop is less that 8% during full load operation.
Following figure presents output characteristic for
speed range from 60 to 180 rpm.
9. Conclusions
This paper has described the most important steps
of designing magnetic circuit of coreless PMSM.
Detailed analytical analysis has been made and
two-dimensional FEM analysis has been applied in
order to present flux distribution. The best size of
the air-gap has been selected on the basis of output
power calculation. The air-gap should be equal to
160% – 170% of the magnet thickness. Coil
diameter and thickness has also been discussed.
Flux distribution, which has been showed in FEM
analysis, has told that the magnet cannot be
widened up to 100% of pole pitch because of
leakage flux. Trapezoidal magnet shape has been
proposed in order to increase output voltage
because of trapezoidal shape of flux density.
Proper distance between magnets should be
designed in this case. For verification, advanced
prototype has been made. Machine has no cogging
torque and starting torque is less than 2 Nm.
Symmetrical three-phase windings provide
constant torque at full range of power. Thanks to
the coreless technology there is no hysteresis and
eddy currents looses. That is why, achieved
efficiency is 96% at 250 rpm and can be increased
up to 98% at 600 rpm. During this operation
machine can provide up to 20kW of active power.
The last part of this paper describes experimental
results. There is shown output characteristic of
PMSG.
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Acknowledgement
This work was financially supported by Institute of
Control and Industrial Electronics, Warsaw
University of Technology, Warsaw, Poland.
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